Prerequisite: None
The Power Rule is simple. If f(x) = Cxn, then f'(x) = Cnxn - 1.
If f(x) = 5x^2, find f'(3).
This is a relatively simple question. The power rule can be used on this function.
f'(x) = 10x
Now we need to evaulate f'(3).
f'(3) = 10(3)
= 30
The correct response would be 30. But this was a single term question. Let us ramp it up a little bit.
If f(x) = x^3 - 9x, find f'(-4).
A little more complicated than the previous question. Differenation is done on a per term basis.
f'(x) = 3x^2 - 9
Now we need to evaulate f'(-4).
f'(-4) = 3(-4)^2 - 9
= 3(16) - 9
= 48 - 9
= 39
The correct response would be 39.
There is a dedicated practice module available. The goal is to score 25 points, and there is also a safety net of 3 questions. (No life is deducted for wrong answers until the safety net wears off.)
Prerequisite: None
Trigonometry has its own set of derivatives any daring individual must know. These are required to conquer Arcade mode.
| Base function | Derivative |
|---|---|
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec2(x) |
| csc(x | -csc(x)*cot(x) |
| sec(x) | sec(x)*tan(x) |
| cot(x) | -csc2(x) |
There is a dedicated practice module available. With the assistance of a graphing calculator, the goal is to score 12 points, and there is no safety net. (Life will be immediately deducted with each wrong answer.)
Prerequisite: Mastery of the Power Rule
The Product Rule is another important rule in differentiation. If we got f(x)*g(x), then the derivative is f'(x)g(x) + f(x)g'(x).
Note: This specific question is not in the question pool itself, but another question testing this concept is in the pool.
Differentiate (x^3 + 2x + 8)(x^3 - 4x^2 + 8).
This question is very difficult, but still solvable. We could try to do all this multiplication manually, but it would be painful. Instead, lets use our new rule.
f'(x) = (3x^2 + 2)(x^3 - 4x^2 + 8) + (x^3 + 2x + 8)(3x^2 - 8x)
Slightly less painful, but still difficult.
f'(x) = 3x^5 - 12x^4 + 24x^2 + 2x^3 - 8x^2 + 16 + 3x^5 - 8x^4 + 6x^3 - 16x^2 + 24x^2 - 64x
Heavy use of the Distributive property makes this look very ugly.
f'(x) = 6x^5 - 20x^4 + 8x^3 + 24x^2 - 64x + 16
There, all the mess has been cleaned up. The correct, and most simplified, answer is 6x5 - 20x4 + 8x3 + 24x2 - 64x + 16.
Prerequisite: Mastery of the Power Rule
The Quotient Rule involves a function that has a fraction containing a sub-function on each side. If we were to differentiate f(x)/g(x), then the result would be [f'(x)g(x) - f(x)g'(x)]/[g(x)]2.
Note: This specific question is not in the question pool itself, but another question testing this concept is in the pool.
Differentiate (x^2 + 4)/(x^4 + 2x^2 + 4).
As usual, we've got a complicated question. This time, we need the Quotient Rule to save us.
f'(x) = [(2x)(x^4 + 2x^2 + 4) - (x^2 + 4)(4x^3 + 4x)]/[(x^4 + 2x^2 + 4)]^2
Let's try reducing this mess.
f'(x) = [(2x^5 + 4x^3 + 8x) - (4x^5 + 20x^3 + 16x)]/[(x^4 + 2x^2 + 4)]^2
Looks better, but could still be improved upon.
f'(x) = (-2x^5 - 16x^3 - 8x)/[(x^4 + 2x^2 + 4)]^2
This concludes the question. The divisor could still be expanded upon, but that is not necessary to understand this rule. Still, we will leave a spoiler anyway.
Spoiler: (-2x5 - 16x3 - 8x)/(x8 + 4x6 + 12x4 + 16x2 + 16)
Prerequisite: General mastery of differentiation
The Chain Rule can be very tricky to master. You start with the outside and work your way in.
Differentiating f(g(x)) would yield f'(g(x))*g'(x).Differentiate (x^2 + 1)^3.
This definitely looks complicated. Lets start with the outside. Differentiating u3 = 3u2*du.
f'(x) = 3(x^2 + 1)^2 * ???
Now for the inside. Differentiating x2 + 1 = 2x.
f'(x) = 3(x^2 + 1)^2 * 2x
= 6x(x^2 + 1)^2
The correct response would be 6x(x2 + 1)2.
Though not accepted by Differenation Challenge; if you truly wanted to, you could still expand the answer further... The expanded answer is spoilered to allow for checking work.
Spoiler: 24x5 + 12x3 + 6x
Prerequisite: General mastery of differentiation
Do not allow word problems to throw you off. They serve as applications of differentiation.
A ball's height is modeled by h(t) = 1327 - 16t^2.
At t = 7, what is the velocity of the ball?
The derivative of position (e.g. change in position) would be the velocity.
v(t) = h'(t)
= -32t
v(7) = -32(7)
= -224
The correct answer would be -224.